Document discretionary_policy_engine.m
Johannes Pfeifer
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@ -1,16 +1,54 @@
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function [H,G,retcode]=discretionary_policy_engine(AAlag,AA0,AAlead,BB,bigw,instr_id,beta,solve_maxit,discretion_tol,qz_criterium,H00,verbose)
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% Solves the discretionary problem for a model of the form:
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% AAlag*yy_{t-1}+AA0*yy_t+AAlead*yy_{t+1}+BB*e=0, with W the weight on the
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% variables in vector y_t and instr_id is the location of the instruments
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% in the yy_t vector.
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%
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% Loss=E_0 sum_{t=0}^{\infty} beta^t [y_t'*W*y+x_t'*Q*x_t]
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% subject to
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% AAlag*yy_{t-1}+AA0*yy_t+AAlead*yy_{t+1}+BB*e=0
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%
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% with W the weight on the variables in vector y_t.
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%
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% The solution takes the form
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% y_t=H*y_{t-1}+G*e_t
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% where H=[H1;F1] and G=[H2;F2].
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%
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% We use the Dennis (2007, Macroeconomic Dynamics) algorithm and so we need
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% to re-write the model in the form
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% A0*y_t=A1*y_{t-1}+A2*y_{t+1}+A3*x_t+A4*x_{t+1}+A5*e_t, with W the
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% weight on the y_t vector and Q the weight on the x_t vector of
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% instruments.
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%
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% Inputs:
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% AAlag [double] matrix of coefficients on lagged
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% variables
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% AA0 [double] matrix of coefficients on
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% contemporaneous variables
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% AAlead [double] matrix of coefficients on
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% leaded variables
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% BB [double] matrix of coefficients on
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% shocks
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% bigw [double] matrix of coefficients on variables in
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% loss/objective function; stacks [W and Q]
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% instr_id [double] location vector of the instruments in the yy_t vector.
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% beta [scalar] planner discount factor
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% solve_maxit [scalar] maximum number of iterations
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% discretion_tol [scalar] convergence criterion for solution
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% qz_criterium [scalar] tolerance for QZ decomposition
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% H00
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% verbose [scalar] dummy to control verbosity
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%
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% Outputs:
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% H [double] (endo_nbr*endo_nbr) solution matrix for endogenous
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% variables, stacks [H1 and H1]
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% G [double] (endo_nbr*exo_nbr) solution matrix for shocks, stacks [H2 and F2]
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%
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% retcode [scalar] return code
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%
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% Algorithm:
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% Dennis, Richard (2007): Optimal policy in rational expectations models: new solution algorithms,
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% Macroeconomic Dynamics, 11, 31–55.
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% Copyright (C) 2007-2012 Dynare Team
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% Copyright (C) 2007-2015 Dynare Team
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%
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% This file is part of Dynare.
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%
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@ -53,14 +91,15 @@ end
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[A0,A1,A2,A3,A4,A5,W,Q,endo_nbr,exo_nbr,aux,endo_augm_id]=GetDennisMatrices(AAlag,AA0,AAlead,BB,bigw,instr_id);
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% aux is a logical index of the instruments which appear with lags in the
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% model. Their location in the state vector is instr_id(aux)
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% model. Their location in the state vector is instr_id(aux);
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% endo_augm_id is index (not logical) of locations of the augmented vector
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% of non-instrumental variables
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AuxiliaryVariables_nbr=sum(aux);
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H0=zeros(endo_nbr+AuxiliaryVariables_nbr);
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if ~isempty(H00)
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H0(1:endo_nbr,1:endo_nbr)=H00;clear H00
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H0(1:endo_nbr,1:endo_nbr)=H00;
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clear H00
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end
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H10=H0(endo_augm_id,endo_augm_id);
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@ -69,6 +108,7 @@ F10=H0(instr_id,endo_augm_id);
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iter=0;
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H1=H10;
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F1=F10;
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%solve equations (20) and (22) via fixed point iteration
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while 1
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iter=iter+1;
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P=SylvesterDoubling(W+beta*F1'*Q*F1,beta*H1',H1,discretion_tol,solve_maxit);
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@ -79,11 +119,11 @@ while 1
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return
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end
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end
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D=A0-A2*H1-A4*F1;
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D=A0-A2*H1-A4*F1; %equation (20)
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Dinv=inv(D);
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A3DPD=A3'*Dinv'*P*Dinv;
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F1=-(Q+A3DPD*A3)\(A3DPD*A1);
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H1=Dinv*(A1+A3*F1);
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A3DPD=A3'*Dinv'*P*Dinv;
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F1=-(Q+A3DPD*A3)\(A3DPD*A1); %component of (26)
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H1=Dinv*(A1+A3*F1); %component of (27)
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[rcode,NQ]=CheckConvergence([H1;F1]-[H10;F10],iter,solve_maxit,discretion_tol);
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if rcode
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@ -97,16 +137,17 @@ while 1
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F10=F1;
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end
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%check if successful
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retcode = 0;
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switch rcode
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case 3 % nan
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retcode=63;
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retcode(2)=10000;
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if verbose
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disp([mfilename,':: NAN elements in the solution'])
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disp([mfilename,':: NaN elements in the solution'])
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end
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case 2% maxiter
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retcode = 61
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retcode = 61;
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if verbose
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disp([mfilename,':: Maximum Number of Iterations reached'])
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end
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@ -125,8 +166,8 @@ if retcode(1)
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H=[];
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G=[];
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else
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F2=-(Q+A3DPD*A3)\(A3DPD*A5);
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H2=Dinv*(A5+A3*F2);
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F2=-(Q+A3DPD*A3)\(A3DPD*A5); %equation (29)
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H2=Dinv*(A5+A3*F2); %equation (31)
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H=zeros(endo_nbr+AuxiliaryVariables_nbr);
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G=zeros(endo_nbr+AuxiliaryVariables_nbr,exo_nbr);
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H(endo_augm_id,endo_augm_id)=H1;
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@ -159,6 +200,7 @@ end
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end
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function [A00,A11,A22,A33,A44,A55,WW,Q,endo_nbr,exo_nbr,aux,endo_augm_id]=GetDennisMatrices(AAlag,AA0,AAlead,BB,bigw,instr_id)
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%get the matrices to use the Dennis (2007) algorithm
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[eq_nbr,endo_nbr]=size(AAlag);
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exo_nbr=size(BB,2);
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y=setdiff(1:endo_nbr,instr_id);
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@ -211,7 +253,7 @@ end
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function v = SylvesterHessenbergSchur(d,g,h)
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%
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% DSYLHS Solves a discrete time sylvester equation using the
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% DSYLHS Solves a discrete time sylvester equation using the
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% Hessenberg-Schur algorithm
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%
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% v = DSYLHS(g,d,h) computes the matrix v that satisfies the
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